PART I. This will be considered as your P1 Quiz No. 1: 1. \( \int 12 t^{7}-t^{2}-t+2 d t \) 2. \( \int 10 w^{4}+9 w^{3}+7 w d w \) 3. \( \int_{0}^{2}\left(3 x^{3}-7 x\right) d x \) 4. \( \int_{2}^{4} \frac{x+1}{x^{2}+2 x+1} d x \) 5. \( \int \frac{x^{4}+3 x^{3}+2 x+6}{x+3} d x \) 6. Determine \( f(x) \), given that \( f^{\prime}(x)=6 x^{8}-20 x^{4}+x^{2}+9 \). PART II. This will be considered as your P1 Quiz No. 2: 1. \( \int \frac{e^{x}}{e^{x}+1} d x \) 2. \( \int b x e^{a x^{2}+1} d x \), wherein \( \mathrm{a} \& \mathrm{~b} \) are constants. 3. \( \int \frac{e^{\ln (1-t)}}{1-t} d t \) 4. \( \int_{2}^{4} \frac{1+2 x+x^{2}}{3 x+3 x^{2}+x^{3}} d x \) 5. \( \int \frac{d x}{x(\ln x)^{2}} \) 6. \( \int \tan x d x \)

Evaluate the integral by following steps: - step0: Solution: \(\int \frac{e^{\ln{\left(1-t\right)}}}{1-t} dt\) - step1: Transform the expression: \(\int \frac{1-t}{1-t} dt\) - step2: Divide the terms: \(\int 1 dt\) - step3: Evaluate the integral: \(t\) - step4: Add the constant of integral C: \(t + C, C \in \mathbb{R}\) Calculate the integral \( \frac{x+1}{x^{2}+2 x+1} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{x+1}{x^{2}+2x+1} dx\) - step1: Simplify the expression: \(\int \frac{1}{x+1} dx\) - step2: Evaluate the integral: \(\ln{\left(\left|x+1\right|\right)}\) - step3: Add the constant of integral C: \(\ln{\left(\left|x+1\right|\right)} + C, C \in \mathbb{R}\) Calculate the integral \( \tan x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \tan\left(x\right) dx\) - step1: Evaluate the integral: \(\ln{\left(\left|\sec\left(x\right)\right|\right)}\) - step2: Add the constant of integral C: \(\ln{\left(\left|\sec\left(x\right)\right|\right)} + C, C \in \mathbb{R}\) Calculate the integral \( \frac{x^{4}+3 x^{3}+2 x+6}{x+3} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{x^{4}+3x^{3}+2x+6}{x+3} dx\) - step1: Simplify the expression: \(\int \left(x^{3}+2\right) dx\) - step2: Use properties of integrals: \(\int x^{3} dx+\int 2 dx\) - step3: Evaluate the integral: \(\frac{x^{4}}{4}+\int 2 dx\) - step4: Evaluate the integral: \(\frac{x^{4}}{4}+2x\) - step5: Add the constant of integral C: \(\frac{x^{4}}{4}+2x + C, C \in \mathbb{R}\) Calculate the integral \( \frac{e^{x}}{e^{x}+1} \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{e^{x}}{e^{x}+1} dx\) - step1: Use the substitution \(dx=e^{-x} dt\) to transform the integral\(:\) \(\int \frac{e^{x}}{e^{x}+1}\times e^{-x} dt\) - step2: Simplify: \(\int \frac{1}{e^{x}+1} dt\) - step3: Use the substitution \(t=e^{x}\) to transform the integral\(:\) \(\int \frac{1}{t+1} dt\) - step4: Evaluate the integral: \(\ln{\left(\left|t+1\right|\right)}\) - step5: Substitute back: \(\ln{\left(\left|e^{x}+1\right|\right)}\) - step6: Calculate the absolute value: \(\ln{\left(e^{x}+1\right)}\) - step7: Add the constant of integral C: \(\ln{\left(e^{x}+1\right)} + C, C \in \mathbb{R}\) Calculate the integral \( 3 x^{3}-7 x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(3x^{3}-7x\right) dx\) - step1: Use properties of integrals: \(\int 3x^{3} dx+\int -7x dx\) - step2: Evaluate the integral: \(\frac{3x^{4}}{4}+\int -7x dx\) - step3: Evaluate the integral: \(\frac{3x^{4}}{4}-\frac{7x^{2}}{2}\) - step4: Add the constant of integral C: \(\frac{3x^{4}}{4}-\frac{7x^{2}}{2} + C, C \in \mathbb{R}\) Calculate the integral \( 10 w^{4}+9 w^{3}+7 w \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(10w^{4}+9w^{3}+7w\right) dw\) - step1: Use properties of integrals: \(\int 10w^{4} dw+\int 9w^{3} dw+\int 7w dw\) - step2: Evaluate the integral: \(2w^{5}+\int 9w^{3} dw+\int 7w dw\) - step3: Evaluate the integral: \(2w^{5}+\frac{9w^{4}}{4}+\int 7w dw\) - step4: Evaluate the integral: \(2w^{5}+\frac{9w^{4}}{4}+\frac{7w^{2}}{2}\) - step5: Add the constant of integral C: \(2w^{5}+\frac{9w^{4}}{4}+\frac{7w^{2}}{2} + C, C \in \mathbb{R}\) Calculate the integral \( \frac{1}{x(\ln x)^{2}} \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{1}{x\left(\ln{\left(x\right)}\right)^{2}} dx\) - step1: Simplify the expression: \(\int \frac{1}{\left(\ln{\left(x\right)}\right)^{2}x} dx\) - step2: Use the substitution \(dx=x dt\) to transform the integral\(:\) \(\int \frac{1}{\left(\ln{\left(x\right)}\right)^{2}x}\times x dt\) - step3: Simplify: \(\int \frac{1}{\left(\ln{\left(x\right)}\right)^{2}} dt\) - step4: Use the substitution \(t=\ln{\left(x\right)}\) to transform the integral\(:\) \(\int \frac{1}{t^{2}} dt\) - step5: Evaluate the integral: \(\frac{t^{-2+1}}{-2+1}\) - step6: Add the numbers: \(\frac{t^{-1}}{-2+1}\) - step7: Add the numbers: \(\frac{t^{-1}}{-1}\) - step8: Divide the terms: \(-t^{-1}\) - step9: Express with a positive exponent: \(-\frac{1}{t}\) - step10: Substitute back: \(-\frac{1}{\ln{\left(x\right)}}\) - step11: Add the constant of integral C: \(-\frac{1}{\ln{\left(x\right)}} + C, C \in \mathbb{R}\) Calculate the integral \( 6 x^{8}-20 x^{4}+x^{2}+9 \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(6x^{8}-20x^{4}+x^{2}+9\right) dx\) - step1: Use properties of integrals: \(\int 6x^{8} dx+\int -20x^{4} dx+\int x^{2} dx+\int 9 dx\) - step2: Evaluate the integral: \(\frac{2x^{9}}{3}+\int -20x^{4} dx+\int x^{2} dx+\int 9 dx\) - step3: Evaluate the integral: \(\frac{2x^{9}}{3}-4x^{5}+\int x^{2} dx+\int 9 dx\) - step4: Evaluate the integral: \(\frac{2x^{9}}{3}-4x^{5}+\frac{x^{3}}{3}+\int 9 dx\) - step5: Evaluate the integral: \(\frac{2x^{9}}{3}-4x^{5}+\frac{x^{3}}{3}+9x\) - step6: Add the constant of integral C: \(\frac{2x^{9}}{3}-4x^{5}+\frac{x^{3}}{3}+9x + C, C \in \mathbb{R}\) Calculate the integral \( 12 t^{7}-t^{2}-t+2 \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(12t^{7}-t^{2}-t+2\right) dt\) - step1: Use properties of integrals: \(\int 12t^{7} dt+\int -t^{2} dt+\int -t dt+\int 2 dt\) - step2: Evaluate the integral: \(\frac{3t^{8}}{2}+\int -t^{2} dt+\int -t dt+\int 2 dt\) - step3: Evaluate the integral: \(\frac{3t^{8}}{2}-\frac{t^{3}}{3}+\int -t dt+\int 2 dt\) - step4: Evaluate the integral: \(\frac{3t^{8}}{2}-\frac{t^{3}}{3}-\frac{t^{2}}{2}+\int 2 dt\) - step5: Evaluate the integral: \(\frac{3t^{8}}{2}-\frac{t^{3}}{3}-\frac{t^{2}}{2}+2t\) - step6: Add the constant of integral C: \(\frac{3t^{8}}{2}-\frac{t^{3}}{3}-\frac{t^{2}}{2}+2t + C, C \in \mathbb{R}\) Calculate the integral \( \frac{1+2 x+x^{2}}{3 x+3 x^{2}+x^{3}} \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{1+2x+x^{2}}{3x+3x^{2}+x^{3}} dx\) - step1: Use the substitution \(dx=\frac{1}{3+6x+3x^{2}} dt\) to transform the integral\(:\) \(\int \frac{1+2x+x^{2}}{3x+3x^{2}+x^{3}}\times \frac{1}{3+6x+3x^{2}} dt\) - step2: Simplify: \(\int \frac{1}{3\left(3x+3x^{2}+x^{3}\right)} dt\) - step3: Use the substitution \(t=3x+3x^{2}+x^{3}\) to transform the integral\(:\) \(\int \frac{1}{3t} dt\) - step4: Rewrite the expression: \(\int \frac{1}{3}\times \frac{1}{t} dt\) - step5: Use properties of integrals: \(\frac{1}{3}\times \int \frac{1}{t} dt\) - step6: Evaluate the integral: \(\frac{1}{3}\ln{\left(\left|t\right|\right)}\) - step7: Substitute back: \(\frac{1}{3}\ln{\left(\left|3x+3x^{2}+x^{3}\right|\right)}\) - step8: Add the constant of integral C: \(\frac{1}{3}\ln{\left(\left|3x+3x^{2}+x^{3}\right|\right)} + C, C \in \mathbb{R}\) ### PART I: P1 Quiz No. 1 Solutions 1. **Integral**: \[ \int (12 t^{7}-t^{2}-t+2) \, dt = \frac{3t^{8}}{2}-\frac{t^{3}}{3}-\frac{t^{2}}{2}+2t + C, \quad C \in \mathbb{R} \] 2. **Integral**: \[ \int (10 w^{4}+9 w^{3}+7 w) \, dw = 2w^{5}+\frac{9w^{4}}{4}+\frac{7w^{2}}{2} + C, \quad C \in \mathbb{R} \] 3. **Definite Integral**: \[ \int_{0}^{2} (3 x^{3}-7 x) \, dx = \frac{3x^{4}}{4}-\frac{7x^{2}}{2} \bigg|_{0}^{2} = \left(\frac{3(2)^{4}}{4}-\frac{7(2)^{2}}{2}\right) - \left(0\right) = 12 - 14 = -2 \] 4. **Definite Integral**: \[ \int_{2}^{4} \frac{x+1}{x^{2}+2 x+1} \, dx = \ln{(|x+1|)} \bigg|_{2}^{4} = \ln{(5)} - \ln{(3)} = \ln{\left(\frac{5}{3}\right)} \] 5. **Integral**: \[ \int \frac{x^{4}+3 x^{3}+2 x+6}{x+3} \, dx = \frac{x^{4}}{4}+2x + C, \quad C \in \mathbb{R} \] 6. **Finding \( f(x) \)**: \[ f(x) = \int (6 x^{8}-20 x^{4}+x^{2}+9) \, dx = \frac{2x^{9}}{3}-4x^{5}+\frac{x^{3}}{3}+9x + C, \quad C \in \mathbb{R} \] --- ### PART II: P1 Quiz No. 2 Solutions 1. **Integral**: \[ \int \frac{e^{x}}{e^{x}+1} \, dx = \ln{(e^{x}+1)} + C, \quad C \in \mathbb{R} \] 2. **Integral**: The expression \( \int b x e^{a x^{2}+1} \, dx \) has a syntax error. Please provide a valid expression. 3. **Integral**: \[ \int \frac{e^{\ln (1-t)}}{1-t} \, dt = t + C, \quad C \in \mathbb{R} \] 4. **Definite Integral**: \[ \int_{2}^{4} \frac{1+2 x+x^{2}}{3 x+3 x^{2}+x^{3}} \, dx = \frac{1}{3}\ln{(|3x+3x^{2}+x^{3}|)} \bigg|_{2}^{4} = \frac{1}{3}(\ln{(75)} - \ln{(30)}) = \frac{1}{3}\ln{\left(\frac{75}{30}\right)} = \frac{1}{3}\ln{\left(\frac{5}{2}\right)} \] 5. **Integral**: \[ \int \frac{1}{x(\ln x)^{2}} \, dx = -\frac{1}{\ln{(x)}} + C, \quad C \in \mathbb{R} \] 6. **Integral**: \[ \int \tan x \, dx = \ln{(|\sec(x)|)} + C, \quad C \in \mathbb{R} \] If you have any further questions or need additional assistance, feel free to ask!